I will chime in with my own experience. First, setting the WB correctly is virtually impossible due to the cycle lighting. Inevitibly, there will be pretty severe color cast on most images while some are perfect, again due to cycling. That being said, shoot in RAW and solve that problem easily in post. Most raw converters have that ability as a standard feature.
Second, make sure that you have a lense that will let in enough light e.g. f 1.8. 4,2. and shoot wide open. I have seen some good results with f 2.8 but I would be willing to bet that there are not as many keepers.
Next, shoot in shutter priority. The first thing I do is set the speed to 1/250 (minimum!) and check the exposure. Obviously too dark. Now bump the iso to 400, still too dark, now to 800, some gyms allow you to stop right there. If it is still too dark go to 1600. I have only had to brave 3200 one time. In a lot of gyms you can get up to 1/500 at 1600.
Now for metering, I always use spot metering and some may suggest matrix.
Now for focusing, I use spot focus, AF-C for floor and AF-S for all other events.
Now for technique. I know a lot of people love the spray and pray approach using burst mode. I personally do not do it that often unless I am trying to capture the sequence of a skill e.g. back handspring, dismounts, etc. I think knowing the routine is very important to you can plan your shots and think about where you should be to get them.
Beam, I think, is the easiest event to shoot that yields the best results. There are many artistic dance moves that can be had. For the skills like a split leap if you focus on catching the girl at the top of a jump you get dramatic images. Here is an example:
(out of focus, I know)
Vault. I find that the best images can be had standing off to the side of the vault track and pre focusing on the vault table. Most vaults require some sort of blocking off of the table so you can get an action shot with good facial expressions. I find that standing behind the vault table and trying to get pics that way is hard do to the lense. I use an 85mm 1.8 and I can never get in the right position to fill the frame. Here are a couple of examples:
Floor, to me, is the hardest to shoot. Hard to get a good backround and usually a lot of obstructions from Judges table, other team, etc.. I really do not spend a lot of time trying to get action shots from tumbling passes. I try to capture interesting compositions like this one.
Candids are what I call the safe shots. There are many to be had. The focus on the gymnast face while she is staring at the judges waiting for the green flag. Coaches interaction with the girls. The interaction between the team. Etc.. This photo tells a story that could only be understood by our gym. One of our girls had a neck injury but was competing though the pain, with the help of her BFF.
Well, there is my 2 cents. I look forward to hearing, and seeing, from others. I am sure GYMDAD will wiegh in on this thread with some good wisdom.